Animating cylinder mesh distortion with bezier curve
You'll be able to do that, adding and tuning "shape keys": https://www.blender.org/manual/animation/shape_keys/index.html
You might find plenty tutorials on shape keys by modifiers, and shape keys by other shapes. After you have your shape keys set up, you just need to keyframe the "evaluation time", and Blender will interpolate the transition between the cilinder, to the "curved cilinder".
By the documentation, a shape key procedure is: 

1.In Object Mode, add a new shape keys via the Shape Key panel with the + button. 2. “Basis” is the rest shape. “Key 1”, “Key 2”, etc.
will be the new shapes.
3. Switch to Edit Mode, select “Key 1” in the Shape Key panel.
4. Deform mesh as you want (do not remove or add vertices).
5. Select “Key 2”, the mesh will be changed to the rest shape. Transform “Key 2” and keep going for other shape keys. Switch back
to Object Mode. Set the Value for “Key 1”, “Key 2”, etc. to see the
transformation between the shape keys.